No, since this is the square root of a negative number. The work done is mgh, and the change of kinetic energy is 1/2 m(v2)^2 - 1/2 m(v1)^2, where v2 is the speed at the height of 15m, and v1 the initial speed. So, this implies 2gh + (v1)^2 = (v2)^2.

Of course, it is clear what you wanted to say, but your equation 1/2mv^2=1/2mv^2+mgh implies 0 = mgh, and there is no 'v2' in it. So, be just a bit more precise when writing something.

Further on, it is not easier to think about this as conservation of energy, it's just another point of view.

I think conservation of energy

PE1+KE1=PE2+KE2

Clearly indicated that there were 2 KE's and therefore 2 velocities. And try not to tell people to be a bit more precise after you have just made a much more careless error. Mine was technical, yours was a mistake.

Furthermore, in introductory physics these concepts are developed through the conservation of energy law-always. I think that any physics teacher would agree that its necessary to understand the laws before trying to link work and PE.

I wouldn't be surprised if the particular chapter he is currently in was titled, "Conservation of Energy"

Think about it this way, the initial energy must be equal to the final energy becasue energy cannot be created or destroyed, it can only change form. This is whats happening here, initially it has KE but no PE (PE is relevent to some reference point, we are choosing the initial reference point to be 0 for simplicity).

Now since it has gone up 15m from the reference 0 it has gianed PE, but where did this PE come from? It cannot be created so it must have come from a change, some of the KE transformed to PE.

Therefore the sum of energy to begin with must equal the sum of energy at the end.

the initial energy

PE+KE1, but at the begining you have no PE to the reference because we have chosen 0 to be the reference (0-0=0)

In which case you end up with a negative number (-694), should be a hint that you have made a mistake because you cannot take the sqrt of a negative number w/o complex numbers. Your algebra MUST be solid before you can tackle physics, no matter how many calculus/physics/engineering classes you ever take, you will find the without solid algebra, you will always struggle.

Everyone makes mistakes, its recognizing them that makes the difference, often the math will tell you that you need to go back and check something like in this case.